Method of generating space-time codes for generalized layered space-time architectures

ABSTRACT

Space-time codes for use with layered architectures with arbitrary numbers of antennas are provided such as rate k/n convolutional codes (e.g., rates higher than or equal to 1/n where n is the number of transmit antennas). Convolutional codes for layered space-time architectures are generated using matrices over the ring F[[D]] of formal power series in variable D.

[0001] This application claims the benefit of provisional U.S.application Serial No. 60/153,936, filed Sep. 15, 1999.

CROSS REFERENCE TO RELATED APPLICATIONS

[0002] Related subject matter is disclosed in U.S. patent applicationSer. No. 09/397,896, filed Sep. 17, 1999, and U.S. patent application ofHesham El Gamal et al for “System Employing Threaded Space-TimeArchitecture for Transporting Symbols and Receivers for Multi-UserDetection and Decoding of Symbols”, filed even date herewith (Attorney'sdocket PD-9900173), the entire contents of both said applications beingexpressly incorporated herein by reference.

FIELD OF THE INVENTION

[0003] The invention relates generally to generating codes for use inlayered space-time architectures.

BACKGROUND OF THE INVENTION

[0004] Unlike the Gaussian channel, the wireless channel suffers frommulti-path fading. In such fading environments, reliable communicationis made possible only through the use of diversity techniques in whichthe receiver is afforded multiple replicas of the transmitted signalunder varying channel conditions.

[0005] Recently, information theoretic studies have shown that spatialdiversity provided by multiple transmit and/or receive antennas allowsfor a significant increase in the capacity of wireless communicationsystems operated in Rayleigh fading environment. Two approaches forexploiting this spatial diversity have been proposed. In the firstapproach, channel coding is performed across the spatial dimension, aswell as the time, to benefit from the spatial diversity provided byusing multiple transmit antennas. The term “space-time codes” is used torefer to this coding scheme. One potential drawback of this schemes isthat the complexity of the maximum likelihood (ML) decoder isexponential in the number of transmit antennas. A second approach reliesupon a layering architecture at the transmitter and signal processing atthe receiver to achieve performance asymptotically close to the outagecapacity. In this “layered” space-time architecture, no attempt is madeto optimize the channel coding scheme. Further, conventional channelcodes are used to minimize complexity. Accordingly, a need exists for alayering architecture, signal processing, and channel coding that aredesigned and optimized jointly.

SUMMARY OF THE INVENTION

[0006] The disadvantages of existing channel coding methods andreceivers for multiple antenna communication systems are overcome and anumber of advantages are realized by the present invention whichprovides space-time codes for use in layered architectures havingarbitrary numbers of antennas and arbitrary constellations. Algebraicdesigns of space-time codes for layered architectures are provided inaccordance with the present invention.

[0007] In accordance with an aspect of the present invention, rate k/nconvolutional codes are provided for layered space-time architectures(e.g., rates higher than or equal to 1/n where n is the number oftransmit antennas).

[0008] In accordance with another aspect of the present invention,convolutional codes for layered space-time architectures are generatedusing matrices over the ring F[[D]] of formal power series in variableD.

BRIEF DESCRIPTION OF THE DRAWINGS

[0009] The various aspects, advantages and novel features of the presentinvention will be more readily comprehended from the following detaileddescription when read in conjunction with the appended drawing, inwhich:

[0010]FIG. 1 is a block diagram of a multiple antenna wirelesscommunication system constructed in accordance with an embodiment of thepresent invention.

[0011] Throughout the drawing figures, like reference numerals will beunderstood to refer to like parts and components.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0012] 1. Space-Time Signaling

[0013]FIG. 1 depicts a multiple antenna communication system 10 with ntransmit antennas 18 and m receive antennas 24. In this system 10, thechannel encoder 20 accepts input from the information source 12 andoutputs a coded stream of higher redundancy suitable for errorcorrection processing at the receiver 16. The encoded output stream ismodulated and distributed among the n antennas via a spatial modulator22. The signal received at each antenna 24 is a superposition of the ntransmitted signals corrupted by additive white Gaussian noise andmultiplicative fading. At the receiver 16, the signal r_(t) ^(j)received by antenna j at time t is given by $\begin{matrix}{r_{t}^{j} = {{\sqrt{E_{s}}{\sum\limits_{i = 1}^{n}{\alpha_{t}^{({ij})}c_{t}^{i}}}} + n_{t}^{j}}} & (1)\end{matrix}$

[0014] where {square root}{square root over (E_(s))} is the energy pertransmitted symbol; α_(t)^((ij))

[0015] is the complex path gain from transmit antenna i to receiveantenna j at time t; c_(t) ^(i) is the symbol transmitted from antenna iat time t; n_(t) ^(j); is the additive white Gaussian noise sample forreceive antenna j at time t. The symbols are selected from a discreteconstellation Ω containing 2^(b) points. The noise samples areindependent samples of zero-mean complex Gaussian random variable withvariance N₀/2 per dimension. The different path gains α_(t)^((ij))

[0016] are assumed to be statistically independent.

[0017] The fading model of primary interest is that of a block flatRayleigh fading process in which the code word encompasses B fadingblocks. The complex fading gains are constant over one fading block butare independent from block to block. The quasi-static fading model is aspecial case of the block fading model in which B=1.

[0018] The system 10 provides not one, but nm, potential communicationlinks between a transmitter 14 and a receiver 16, corresponding to eachdistinct transmit antenna 18/receive antenna 24 pairing. The space-timesystem 10 of the present invention is advantageous because it exploitsthese statistically independent, but mutually interfering, communicationlinks to improve communication performance.

[0019] 2. Generalized Layering

[0020] In a layered space-time approach, the channel encoder 20 iscomposite, and the multiple, independent coded streams are distributedin space-time in layers. The system 10 is advantageous because thelayering architecture and associated signal processing associatedtherewith allows the receiver 16 to efficiently separate the individuallayers from one another and can decode each of the layers effectively.In such schemes, there is no spatial interference among symbolstransmitted within a layer (unlike the conventional space-time codedesign approach). Conventional channel codes can be used while theeffects of spatial interference are addressed in the signal processordesign. While this strategy reduces receiver complexity compared to thenon-layered space-time approach, significant gains are possible withoutundue complexity when the encoding, interleaving, and distribution oftransmitted symbols among different antennas are optimized to maximizespatial diversity, temporal diversity, and coding gain.

[0021] A layer is defined herein as a section of the transmissionresources array (i.e., a two-dimensional representation of all availabletransmission intervals on all antennas) having the property that eachsymbol interval within the section is allocated to at most one antenna.This property ensures that all spatial interference experienced by thelayer comes from outside the layer. A layer has the further structuralproperty that a set of spatial and/or temporal cyclic shifts of thelayer within the transmission resource array provides a partitioning ofthe transmission resource array. This allows for a simple repeated useof the layer pattern for transmission of multiple, independent codedstreams.

[0022] Formally, a layer in an n×l transmission resource array can beidentified by an indexing set L⊂I_(n)×I_(l) having the property that thet-th symbol interval on antenna a belongs to the layer if and only if(a, t)∈L. Then, the formal notion of a layer requires that, if (a, t)∈Land (a′, t′)∈L, then either t≠t′ or a≠a′—i.e., that a is a function oft.

[0023] Now, consider a composite channel encoder γ consisting of nconstituent encoders γ₁, γ₂, . . . , γ_(n) operating on independentinformation streams. Let γ_(i): y^(k) ^(_(i)) →y^(N) ^(_(i)) , so thatk=k₁+k₂+ . . . +k, and N=N₁+N₂+ . . . +N_(n). Then, there is apartitioning u=u₁|u₂| . . . |u_(n) of the composite information vectoru∈y^(k) into a set of disjoint component vectors u_(i), of length k_(i),and a corresponding partitioning γ(u)=γ₁(u₁)|γ₂(u₂)| . . . |γ_(n)(u_(n))of the composite code word γ(u) into a set of constituent code wordsγ_(i)(γ_(i)), of length N_(i). In the layered architecture approach, thespace-time transmitter assigns each of the constituent code wordsγ_(i)(u_(i)) to one of a set of n disjoint layers.

[0024] There is a corresponding decomposition of the spatial formattingfunction that is induced by the layering. Let f_(i) denote the componentspatial formatting function, associated with layer L_(i), which agreeswith the composite spatial formatter f regarding the modulation andformatting of the layer elements but which sets all off-layer elementsto complex zero. Then

[0025] f(γ(u))=f₁(γ₁(u₁))+f₂ (γ₂(u₂))+ . . . +f_(n)(γ_(n)(u_(n))).

[0026] 3. Algebraic Space-Time Code Design

[0027] A space-time code C may be defined as an underlying channel codeC together with a spatial modulator function f that parses the modulatedsymbols among the transmit antennas. It is well known that thefundamental performance parameters for space-time codes are (1)diversity advantage, which describes the exponential decrease of decodederror rate versus signal-to-noise ratio (asymptotic slope of theperformance curve in a log-log scale); and (2) coding advantage whichdoes not affect the asymptotic slope but results in a shift in theperformance curve. These parameters are related to the rank andeigenvalues of certain complex matrices associated with the basebanddifferences between two modulated code words.

[0028] Algebraic space-time code designs achieving full spatialdiversity are made possible by the following binary rank criterion forbinary, BPSK-modulated space-time codes:

[0029] Theorem 1 (Binary Rank Criterion) Let C be a linear n×lspace-time code with underlying binary code C of length N=nl where l≧n.Suppose that every non-zero code word ĉ is a matrix of full rank overthe binary field

. Then, for BPSK transmission over the quasi-static fading channel, thespace-time code C achieves full spatial diversity nm.

[0030] Proof: The proof is discussed in the above-referenced applicationSer. No. 09/397,896. □

[0031] Using the binary rank criterion, algebraic construction forspace-time codes is as follows.

[0032] Theorem 2 (Stacking Construction) Let M₁, M₂, M_(n) be binarymatrices of dimension k×l, l≧k, and let C be the n×l space-time code ofdimension k consisting of the code word matrices${\hat{c} = \begin{bmatrix}{\underset{\_}{x}M_{1}} \\{\underset{\_}{x}M_{2}} \\\vdots \\{\underset{\_}{x}M_{n}}\end{bmatrix}}\quad,$

[0033] where x denotes an arbitrary k-tuple of information bits and n≦l.Then C satisfies the binary rank criterion, and thus, for BPSKtransmission over the quasi-static fading channel, achieves full spatialdiversity nm, if and only if M₁, M₂, . . . , M_(n) have the propertythat

[0034] ∀a₁, a₂, . . . , a_(n) ∈

:

[0035] M=a₁M₁⊕a₂M₂⊕ . . . ⊕a_(n)M_(n) is of full rank k unless a₁=a₂a_(n)=0.

[0036] Proof: The proof is discussed in the above-referenced applicationSer. No. 09/397,896. □

[0037] This construction is general for any number of antennas and, whengeneralized, applies to trellis as well as block codes. The BPSKstacking construction and its variations, including a similar versionfor QPSK transmission (in which case the symbol alphabet is

₄, the integers modulo 4), encompass as special cases transmit delaydiversity, hand-crafted trellis codes, rate 1/n convolutional codes, andcertain block and concatenated coding schemes. Especially interesting isthe class of rate 1/n convolutional codes with the optimal d_(free),most of which can be formatted to achieve full spatial diversity.

[0038] In a layered architecture, an even simpler algebraic constructionis applicable to arbitrary signaling constellations. In particular, forthe design of the component space-time code C associated with layer L,we have the following stacking construction using binary matrices forthe quasi-static fading channel.

[0039] Theorem 3 (Generalized Layered Stacking Construction) Let L be alayer of spatial span n. Given binary matrices M₁, M₂, . . . , M_(n) ofdimension k×l, let C be the binary code of dimension k consisting of allcode words of the form g(x)=xM₁|xM₂| . . . xM_(n), where x denotes anarbitrary k-tuple of information bits. Let f_(L) denote the spatialmodulator having the property that the modulated symbols μ(xM_(j))associated with xM_(j) are transmitted in the l/b symbol intervals of Lthat are assigned to antenna j. Then, as the space-time code in acommunication system with n transmit antennas and m receive antennas,the space-time code C consisting of C and f_(L) achieves spatialdiversity dm in a quasi-static fading channel if and only if d is thelargest integer such that M₁, M₂, . . . , M_(n), have the property that

[0040] ∀a₁, a₂, . . . , a_(n) ∈

, a₁+a₂+ . . . +a_(n)=n−d+1:

[0041] M=[a₁M₁a₂M₂ . . . a_(n)M_(n)] is of rank k over the binary field.

[0042] Proof: The proof is discussed in the above-referenced applicationfiled concurrently herewith. □

[0043] Corollary 4 Full spatial diversity nm is achieved if and only ifM₁, M₂, . . . , M_(n) are of rank k over the binary field.

[0044] The natural space-time codes associated with binary, rate 1/n,convolutional codes with periodic bit interleaving are advantageous forthe layered space-time architecture as they can be easily formatted tosatisfy the generalized layered stacking construction. Theseconvolutional codes have been used for a similar application, that is,the block erasure channel. The main advantage of such codes is theavailability of computationally efficient, soft-input/soft-outputdecoding algorithms.

[0045] The prior literature on space-time trellis codes treats only thecase in which the underlying code has rate 1/n matched to the number oftransmit antennas. In the development of generalized layered space-timecode design of the present invention, consider the more general case inwhich the convolutional code has rate greater than 1/n is considered.The treatment includes the case of rate k/n convolutional codesconstructed by puncturing an underlying rate 1/n convolutional code.

[0046] Let C be a binary convolutional code of rate k/n with the usualtransfer function encoder Y(D)=X(D)G(D). In the natural space-timeformatting of C, the output sequence corresponding to Y_(j)(D) isassigned to the j-th transmit antenna. Let F_(l)(D)=[G_(1,l)(D)G_(2,l)(D) . . . G_(n,l)(D)]^(T). Then, the following theorem relatesthe spatial diversity of the natural space-time code associated with Cto the rank of certain matrices over the ring

[[D]] of formal power series in D.

[0047] Theorem 5 Let C denote the generalized layered space-time codeconsisting of the binary convolutional code C, whose k×n transferfunction matrix is G(D)=[F₁(D) F₂(D) . . . F_(n)(D)], and the spatialmodulator f_(L) in which the output Y_(j)(D)=X(D)·F_(j)(D) is assignedto antenna j along layer L. Let v be the smallest integer having theproperty that, whenever a₁+a₂+ . . . +a_(n)=v, the k×n matrix [a₁F₁ a₂F₂. . . a_(n)F_(n)] has full rank k over

. Then the space-time code C achieves d-level spatial transmit diversityover the quasi-static fading channel where d=n−v+1 and v≧k.

[0048] Proof: The proof is discussed in the above-referenced applicationfiled concurrently herewith. □

[0049] Rate 1/n′ convolutional codes with n′<n can also be put into thisframework. This is shown by the following example. Consider the optimald_(free)=5 convolutional code with generators G₀(D)=1+D² andG₁(D)=1+D+D². In the case of two transmit antennas, it is clear that thenatural layered space-time code achieves d=2 level transmit diversity.

[0050] In the case of four transmit antennas, note that the rate 1/2code can be written as a rate 2/4 convolutional code with generatormatrix: ${G(D)} = {\begin{bmatrix}{1 + D} & 0 & {1 + D} & 1 \\0 & {1 + D} & D & {1 + D}\end{bmatrix}\quad.}$

[0051] By inspection, every pair of columns is linearly independent over

[[D]]. Hence, the natural periodic distribution of the code across fourtransmit antennas produces a generalized layered space-time codeachieving the maximum d=3 transmit spatial diversity.

[0052] For six transmit antennas, the code is expressed as a rate 3/6code with generator matrix: ${G(D)} = {\begin{bmatrix}1 & 0 & 1 & 1 & 1 & 1 \\D & 1 & 0 & D & 1 & 1 \\0 & D & 1 & D & D & 1\end{bmatrix}\quad.}$

[0053] Every set of three columns in the generator matrix has full rankover

[[D]], so the natural space-time code achieves maximum d=4 transmitdiversity.

[0054] Thus far, the design of generalized layered space-time codes thatexploit the spatial diversity over quasi-static fading channels has beendiscussed. The results obtained for generalized layered space-time codedesign, however, are easily extended to the more general block fadingchannel. In fact, in the absence of interference from other layers, thequasi-static fading channel under consideration can be viewed as a blockfading channel with receive diversity, where each fading block isrepresented by a different antenna. For the layered architecture with ntransmit antennas and a quasi-static fading channel, there are nindependent and non-interfering fading links per code word that can beexploited for transmit diversity by proper code design. In the case ofthe block fading channel, there is a total of nB such links, where B isthe number of independent fading blocks per code word per antenna. Thus,the problem of block fading code design for the layered architecture isaddressed by simply replacing parameter n by nB.

[0055] For example, the following “multi-stacking construction” is adirect generalization of Theorem 3 to the case of a block fadingchannel. In particular, special cases of the multi-stacking constructionare given by the natural space-time codes associated with rate k/nconvolutional codes in which various arms from the convolutional encoderare assigned to different antennas and fading blocks (in the same waythat Theorem 5 is a specialization of Theorem 3).

[0056] Theorem 6 (Generalized Layered Multi-Stacking Construction) Let Lbe a layer of spatial span n. Given binary matrices M_(1,1), M_(2,1), .. . , M_(n,1), . . . , M_(1,B), . . . , M_(2,B), . . . , M_(n,B) ofdimension k×l, let C be the binary code of dimension k consisting of allcode words of the form

[0057] g(x)=xM_(1,1)|xM_(2,1)| . . . |xM_(n,1)| . . .|xM_(1,B)|xM_(2,B)| . . . |xM_(n,B),

[0058] where x denotes an arbitrary k-tuple of information bits, and Bis the number of independent fading blocks spanning one code word. Letf_(L) denote the spatial modulator having the property that μ(xM_(j,ν))is transmitted in the symbol intervals of L that are assigned to antennaj in the fading block ν.

[0059] Then, as the space-time code in a communication system with ntransmit antennas and m recezve antennas, the space-time code Cconsisting of C and f_(L) achieves spatial diversity dm in a B-blockfading channel if and only if d is the largest integer such thatM_(1,1), M_(2,1), . . . , M_(n,B) have the property that

[0060] ∀a_(1,1), a_(2,1), . . . , a_(n,B) ∈

, a_(1,1)+a_(2,1)+ . . . +a_(n,B)=nB−d+1:

[0061] M=[a_(1,1)M_(1,1)a_(2,1)M_(2,1) . . . , a_(n,B)M_(n,B)] is ofrank k over the binary field.

4. CONCLUSIONS

[0062] An algebraic approach to the design of space-time codes forlayered space-time architectures has been formulated; and new codeconstructions have been presented for the quasi-static fading channel aswell as the more general block fading channel. It is worth noting that,in the absence of interference from other layers, the fading channelexperienced by a given coded layer is equivalent to a block fadingchannel with receive diversity. Thus, the algebraic framework providedby the present invention is also useful for block fading channelswithout transmit diversity.

[0063] Although the present invention has been described with referenceto a preferred embodiment thereof, it will be understood that theinvention is not limited to the details thereof. Various modificationsand substitutions have been suggested in the foregoing description, andothers will occur to those of ordinary skill in the art. All suchsubstitutions are intended to be embraced within the scope of theinvention as defined in the appended claims.

What is claimed is:
 1. A method of generating a space-time code forencoding information symbols comprising the steps of: defining a binaryrank criterion such that C is a linear n×l space-time code withunderlying binary code C of length N=nl where l≧n, and a non-zero codeword c is a matrix of full rank over a binary field

to allow full spatial diversity nm for n transmit antennas and m receiveantennas; and generating binary matrices M₁, M₂, . . . , M_(n) ofdimension k×l, l≧k, C being said n×l space-time code of dimension k andcomprising code word matrices ${\hat{c} = \begin{bmatrix}{\underset{\_}{x}M_{1}} \\{\underset{\_}{x}M_{2}} \\\vdots \\{\underset{\_}{x}M_{n}}\end{bmatrix}}\quad,$

 wherein x denotes an arbitrary k-tuple of said information symbols andn≦l, said binary matrices M₁, M₂, . . . , M_(n) being characterized by∀a₁, a₂, . . . , a_(n) ∈

: M=a₁M₁⊕a₂M₂⊕ . . . ⊕a_(n)M_(n) is of full rank k unless a₁=a₂= . . .=a_(n)=0 to allow said code C to satisfy said binary rank criterion. 2.A method as claimed in claim 1, wherein said generating step is used forbinary phase shift keying transmission over a quasi-static fadingchannel and achieves substantially full spatial diversity nm.
 3. Amethod as claimed in claim 1, wherein said code C is selected from thegroup consisting of a trellis code, a block code, a convolutional code,and a concatenated code.
 4. A method of generating a space-time code forencoding information symbols comprising the steps of: defining L as alayer of spatial span n, and C as a binary code of dimension kcomprising code words having the form g(x)=xM₁|xM₂| . . . |xM_(n), whereM₁, M₂, . . . , M_(n) are binary matrices of dimension k×l and x denotesan arbitrary k-tuple of said information symbols; modulating said codewords xM_(j) such that said modulated symbols μ(xM_(j)) are transmittedin the l/b symbol intervals of L that are assigned to an antenna j (18);and generating a space-time code C comprising C and f_(L) having d bethe largest integer such that M₁, M₂, . . . , M_(n) have the propertythat ∀a₁, a₂, . . . , a_(n) ∈

, a₁+a₂+ . . . +a_(n)=n−d+1: M=[a₁M₁a₂M₂ . . . a_(n)M_(n)] is of rank kover the binary field to achieve spatial diversity dm in a quasi-staticfading channel.
 5. A method as claimed in claim 4, wherein said binarymatrices M₁, M₂, . . . , M_(n) are of rank k over the binary field toachieve substantially full spatial diversity nm for n transmit antennasand m receive antennas.
 6. A method as claimed in claim 4, furthercomprising the step of transmitting said space-time codes at a rateb(n−d+1) bits per signaling interval for a generalized layeredcommunication system having n transmit antennas, a signalingconstellation of size 2^(b), and component codes achieving d-levelspatial transmit diversity constellation.
 7. A method for generating aspace-time code for encoding information symbols comprising the stepsof: defining C as a generalized layered space-time code comprising abinary convolutional code C having a k×n transfer function matrix ofG(D)=[F₁(D) F₂(D) . . . F_(n)(D)], and a spatial modulator f_(L) inwhich the output Y_(j)(D)=X(D)·F_(j)(D) is assigned to an antenna jalong a layer L; defining v as the smallest integer having the propertythat, whenever a₁+a₂+ . . . +a_(n)=v, the k×n matrix [a₁F₁ a₂F₂ . . .a_(n)F_(n) has full rank k over

[[x]]; and generating a space-time code C to achieve d-level spatialtransmit diversity over the quasi-static fading channel where d=n−v+1and v≧k.
 8. A method generating a space-time code for encodinginformation symbols comprising the steps of: defining L as a layer ofspatial span n and C as a binary code of dimension k comprising codewords of the form g(x)=xM_(1,1)|xM_(2,1)| . . . |xM_(n,1)| . . .|xM_(1,B)|xM_(2,B)| . . . |xM_(n,B)  where M_(1,1), M_(2,1), . . . ,M_(n,1), . . . , M_(1,B), M_(n,B) are binary matrices of dimension k×land x denotes an arbitrary k-tuple of information bits, and B as thenumber of independent fading blocks spanning one code word; modulatingsaid code words xM_(j,ν) such that said modulated symbols μ(xM_(j,ν))are transmitted in the symbol intervals of L that are assigned toantenna j in a fading block ν; and generating said space-time code in acommunication system with n transmit antennas and m receive antennassuch that C comprising C and f_(L) achieves spatial diversity dm in aB-block fading channel, d being the largest integer such that M_(1,1),M_(2,1), . . . , M_(n,B) have the property that ∀a_(1,1), a_(2,1), . . ., a_(n,B) ∈

, a_(1,1)+a_(2,1)+ . . . +a_(n,B)=nB−d+1:M=[a_(1,1)M_(1,1)a_(2,1)M_(2,1) . . . a_(n,B)M_(n,B)] is of rank k overthe binary field.
 9. An encoding apparatus in a layered space-timearchitecture comprising: an input device for receiving informationsymbols; and a processing device for encoding said information symbolsusing a space-time code generated by defining a binary rank criterionsuch that C is a linear n×l space-time code with underlying binary codeC of length N=nl where l≧n and a non-zero code word ĉ is a matrix offull rank over a binary field

to allow full spatial diversity nm for n transmit antennas and m receiveantennas, and by generating binary matrices M₁, M₂, . . . , M_(n) ofdimension k×l, l≧k, C being said n×l space-time code of dimension k andcomprising code word matrices ${\hat{c} = \begin{bmatrix}{\underset{\_}{x}M_{1}} \\{\underset{\_}{x}M_{2}} \\\vdots \\{\underset{\_}{x}M_{n}}\end{bmatrix}}\quad,$

 wherein x denotes an arbitrary k-tuple of said information symbols andn≦l, said binary matrices M₁, M₂, . . . , M_(n) being characterized by∀a₁, a₂ . . . , a_(n) ∈

: M=a₁M₁⊕a₂M₂⊕ . . . ⊕a_(n)M_(n) is of full rank k unless a₁=a₂= . . .=a_(n)=0 to allow said code C to satisfy said binary rank criterion. 10.An encoding apparatus as claimed in claim 9, wherein said code C isselected from the group consisting of a trellis code, a block code, aconvolutional code, and a concatenated code.
 11. An encoding apparatusin a layered space-time architecture comprising: an input device forreceiving information symbols; and a processing device for encoding saidinformation symbols using a space-time code generated by defining L as alayer of spatial span n and C as a binary code of dimension k comprisingcode words having the form g(x)=xM₁|xM₂| . . . |xM_(n),where M₁, M₂, . .. , M_(n) are binary matrices of dimension k×l and x denotes anarbitrary k-tuple of said information symbols, by modulating said codewords xM_(j) such that said modulated symbols μ(xM_(j)) are transmittedin the l/b symbol intervals of L that are assigned to an antenna j, andby generating said space-time code C comprising C and f_(L) wherein d isthe largest integer such that M₁, M₂, . . . , M_(n) have the propertythat ∀a₁, a₂, . . . , a_(n) ∈

, a₁+a₂+ . . . +a_(n)=n−d+1: M=[a₁M₁a₂M₂ . . . a_(n)M_(n)] is of rank kover the binary field to achieve spatial diversity dm in a quasi-staticfading channel.
 12. An encoding apparatus as claimed in claim 11,wherein said binary matrices M₁, M₂, . . . , M_(n) are of rank k overthe binary field to achieve substantially full spatial diversity nm forn transmit antennas and m receive antennas.
 13. An encoding apparatus ina layered space-time architecture comprising: an input device forreceiving information symbols; and a processing device for encoding saidinformation symbols using a space-time code generated by defining C as ageneralized layered space-time code comprising a binary convolutionalcode C having a k×n transfer function matrix of G(D)=F₁(D) F₂(D) . . .F_(n)(D)], and by spatial modulating f_(L) in which the outputY_(j)(D)=X(D)·F_(j)(D) is assigned to an antenna j along a layer L, vbeing defined as the smallest integer having the property that, whenevera₁+a₂+ . . . +a_(n)=v, the k×n matrix [a₁F₁ a₂F₂ . . . a_(n)F_(n)] hasfull rank k over

[[x]], and generating said space-time code C achieving d-level spatialtransmit diversity over the quasi-static fading channel where d=n−v+1and v≧k.
 14. An encoding apparatus in a layered space-time architecturecomprising: an input device for receiving information symbols; and aprocessing device for encoding said information symbols using aspace-time code generated by defining L as a layer of spatial span n andC as a binary code of dimension k comprising code words of the formg(x)=xM_(1,1)|xM_(2,1)| . . . |xM_(n,1)| . . . |xM_(1,B)|xM_(2,B)| . . .|xM_(n,B) where M_(1,1), M_(2,1), . . . , M_(n,1), . . . , M_(1,B),M_(2,B), . . . , M_(n,B) are binary matrices of dimension k×l and xdenotes an arbitrary k-tuple of information bits, and B as the number ofindependent fading blocks spanning one code word, modulating said codewords xM_(j,ν) such that said modulated symbols μ(xM_(j,ν)) aretransmitted in the symbol intervals of L that are assigned to antenna jin a fading block ν, and generating said space-time code in acommunication system with n transmit antennas and m receive antennassuch that C comprising C and f_(L) achieves spatial diversity dm in aB-block fading channel, d being the largest integer such that M_(1,1),M_(2,1), . . . , M_(n,B) have the property that ∀a_(1,1), a_(2,1), . . ., a_(n,B) ∈

, a_(1,1)+a_(2,1)+ . . . +a_(n,B)=nB−d+1:M=[a_(1,1)M_(1,1)a_(2,1)M_(2,1) . . . , a_(n,B)M_(n,B)] is of rank kover the binary field.